To solve this problem, we are given the following conditions:
1. The first term of an arithmetic sequence is 6.
2. The common difference is -3.
3. We want to find the explicit form of the arithmetic function [tex]\( f(g) \)[/tex] and determine the seventh term.
Let's break down the steps to find the explicit form of the arithmetic function and then compute the seventh term.
### Step 1: Explicit Form of the Arithmetic Function
The general formula for the [tex]\( g \)[/tex]-th term of an arithmetic sequence is given by:
[tex]\[ a_g = a_1 + (g - 1)d \][/tex]
where:
- [tex]\( a_g \)[/tex] is the [tex]\( g \)[/tex]-th term,
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( d \)[/tex] is the common difference, and
- [tex]\( g \)[/tex] is the term number we are interested in.
Given:
- [tex]\( a_1 = 6 \)[/tex]
- [tex]\( d = -3 \)[/tex]
The explicit form of the arithmetic function ([tex]\( f(g) \)[/tex]) is therefore:
[tex]\[ f(g) = 6 + (g - 1)(-3) \][/tex]
### Step 2: Simplify the Function
Simplify the expression for the explicit form:
[tex]\[ f(g) = 6 + (g - 1)(-3) \][/tex]
[tex]\[ f(g) = 6 - 3(g - 1) \][/tex]
[tex]\[ f(g) = 6 - 3g + 3 \][/tex]
[tex]\[ f(g) = 9 - 3g \][/tex]
So, the explicit form of the arithmetic function [tex]\( f(g) \)[/tex] is:
[tex]\[ f(g) = 9 - 3g \][/tex]
### Step 3: Find the Seventh Term
To find the seventh term ([tex]\( g = 7 \)[/tex]), substitute [tex]\( g = 7 \)[/tex] into the function:
[tex]\[ f(7) = 9 - 3(7) \][/tex]
[tex]\[ f(7) = 9 - 21 \][/tex]
[tex]\[ f(7) = -12 \][/tex]
### Conclusion
The explicit form of the arithmetic function is [tex]\( f(g) = 9 - 3g \)[/tex], and the seventh term is:
[tex]\[ f(7) = -12 \][/tex]
Therefore, the correct choice is:
[tex]\[ f(g) = -3(g - 1) + 6 \][/tex]
with the seventh term being [tex]\(-12\)[/tex].
